Beitr\"age zur Algebra und Geometrie
Contributions to Algebra and Geometry
Volume 36 (1995), No. 2
B. Sturmfels, R. Weismantel and G.M. Ziegler:
Gr\"obner Bases of Lattices
Abstract
There are very close connections between the arithmetic of integer lattices,
algebraic properties of the associated ideals,
and the geometry and the combinatorics of corresponding polyhedra.
In this paper we investigate the generating sets (``Gr\"obner bases'') of
integer lattices that correspond to the Gr\"obner bases of the
associated binomial ideals. Extending results by
Sturmfels \& Thomas,
we obtain a geometric characterization of the universal Gr\"obner basis
in terms of the vertices and edges of the associated corner polyhedra.
In the special case where the lattice has finite index,
the corner polyhedra were studied by Gomory, %% \cite{Gom},
and there is a close connection to the ``group problem in integer
programming.''
We present exponential lower and
upper bounds for the maximal size of a reduced Gr\"obner basis.
The initial complex of (the ideal of) a lattice is shown to be
dual to the boundary of a certain simple polyhedron.